What is the philosophy behind the program? What are the successes? I read Granville's survey (not very carefully, admittedly) and came up with the impression that they are just reproving well known theorems in different formalism. I am sure I am mistaken which is why I am too embarrassed to ask this on MO. Any insight will be truly appreciated!

# Pretentious number theory

What is the philosophy behind the program? What are the successes? I read Granville's survey (not very carefully, admittedly) and came up with the impression that they are just reproving well known theorems in different formalism. I am sure I am mistaken which is why I am too embarrassed to ask this on MO. Any insight will be truly appreciated!

I think Granville was happy with the pretentious approach just giving more intuitive proofs for classical facts (which it does do, at least in my experience). However, Sound really wanted the ideas to lead to new results, and there are now many examples of results first proven by pretentious methods. Terry Tao, Dimitris Koukoulopoulos, Adam Harper, and Sacha Mangerel have some results which use these ideas (perhaps most famously, Tao's resolution of the discrepancy conjecture).

I think the first really big result proven using pretentious methods was Soundararajan's weak subconvexity published in Annals circa 2010.

I think Granville was happy with the pretentious approach just giving more intuitive proofs for classical facts (which it does do, at least in my experience).

Do you really claim that a pretentious proof of PNT is more intuitive than the standard proof?

Not necessarily. But these ideas tell you things like that the classical zero-free region is morally equivalent to saying $\mu \left(n\right)$ doesn't pretend to be the ${n}^{-it}$, and that a Siegel zero for $L(s,{\chi}_{d})$ is essentially $\mu \left(n\right)$ pretending to be ${\chi}_{d}\left(n\right)$ for some real quadratic character ${\chi}_{d}$, which are both ideas I find way more intuitive than the classical formulation of these concepts.

[...]

Do you really claim that a pretentious proof of PNT is more intuitive than the standard proof?

Not necessarily. But these ideas tell you things like that the classical zero-free region is morally equivalent to saying $\mu \left(n\right)$ doesn't pretend to be the ${n}^{-it}$, and that a Siegel zero for $L(s,{\chi}_{d})$ is essentially $\mu \left(n\right)$ pretending to be ${\chi}_{d}\left(n\right)$ for some real quadratic character ${\chi}_{d}$, which are both ideas I find way more intuitive than the classical formulation of these concepts.

Sure. They are somehow the way one should think about things, but arguably not the way one should actually prove things.

[...]

Not necessarily. But these ideas tell you things like that the classical zero-free region is morally equivalent to saying $\mu \left(n\right)$ doesn't pretend to be the ${n}^{-it}$, and that a Siegel zero for $L(s,{\chi}_{d})$ is essentially $\mu \left(n\right)$ pretending to be ${\chi}_{d}\left(n\right)$ for some real quadratic character ${\chi}_{d}$, which are both ideas I find way more intuitive than the classical formulation of these concepts.

Sure. They are somehow the way one should think about things, but arguably not the way one should actually prove things.

Yeah, I agree with that.

Let's also not forget Halasz did good work in this direction long before Granville (and the name "pretentious", which, incidentally, was suggested to Granville by Friedlander, as a way to pull Granville's leg).

Granville acknowledges Halasz. When he gave a talk about this stuff at Princeton ten or so years ago he started it by talking about Halasz's theorem and spent a good chunk of the time saying how visionary it was.

Let's also not forget Halasz did good work in this direction long before Granville (and the name "pretentious", which, incidentally, was suggested to Granville by Friedlander, as a way to pull Granville's leg).

Granville acknowledges Halasz. When he gave a talk about this stuff at Princeton ten or so years ago he started it by talking about Halasz's theorem and spent a good chunk of the time saying how visionary it was.

I guess zbuj wasn't suggesting that Granville doesn't acknowledging Halasz; but rather that it is common these days to say "pretentious number theory as developed by Granville-Soundararajan" when using these ideas, and not acknowledge Halasz at all, unless Halasz's theorem is explicitly being used. Perhaps one should start calling the pretentious pseudometric after Halasz to commemorate his contributions.

Let's also not forget Halasz did good work in this direction long before Granville (and the name "pretentious", which, incidentally, was suggested to Granville by Friedlander, as a way to pull Granville's leg).

Thanks. This answers the most obvious question to an outsider.

Thanks. This answers the most obvious question to an outsider.

If you are not being sarcastic, the "real" meaning of pretentiousness comes from "pretending to be ${n}^{it}$"

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Thanks. This answers the most obvious question to an outsider.

If you are not being sarcastic, the "real" meaning of pretentiousness comes from "pretending to be ${n}^{it}$"

That's just the excuse. Granville has a habit of such things -- he's been trying to make Landau-Selberg-Delange happen for something that's been called Selberg-Delange for decades, just so he can write papers on the LSD method.

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